The word is a result of the heavy Danish influence on the English language. Most people do not realize that the English language roots are really Danish or Jutland. Equiped with this knowledge this word is easy to decipher.

The Danish word for Thanks is tak. In Scotland and upper England it was common to drop the k at the end because of the way words were pronounced during the time of old English and Middle English. Hence the slang word "Ta" which should actually be pronounced "TA-k" but over time became "Ta" is really Tak meaning "Thanks"

Person One: I will give you a ride to the post office to get your check.

Ta is the name of one of the main characters in the literary masterpiece Annette by author extraordinaire Martin Eden. In the story Ta who is a crippled army general from the African country of Toga falls in love with a woman from West Virginia named Annette. See: Annette